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Non-liftable varieties via etale cohomology rings

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Pith's one-line read This paper constructs the first simply-connected smooth projective variety whose ℓ-adic étale cohomology ring admits no Q-form, giving a new obstruction to lifting to characteristic zero.

desk verdict Clever cohomological encoding, but the construction only works over an extension of F_p; Theorem 0.1 as stated is not established. read the letter →

arxiv 2607.18588 v1 pith:B4NTABTI submitted 2026-07-20 math.AG

Runjie Hu , Siqing Zhang This is my paper
classification math.AG MSC 14F2014G17
keywords non-liftablevarietyétalecohomologyringQ-formquaterniondivisionalgebrasupersingularellipticcurveblow-uprank-droplocussimplyconnected
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper constructs a smooth projective variety over a finite field whose ℓ-adic étale cohomology ring cannot be obtained from a graded Q-algebra by extending scalars to Q_ℓ. Since the cohomology ring of any variety that lifts to characteristic zero always has such a Q-form, this yields a new type of non-liftability obstruction. The obstruction is cohomological rather than geometric: it encodes a quaternion division algebra inside the ring itself, rather than relying on pathologies like failure of Kodaira vanishing. The construction uses a supersingular elliptic curve and blow-ups to make the quaternion algebra visible in the cohomology ring, then a further blow-up makes the variety simply connected while preserving the obstruction. This is claimed to be the first example realizing a topological obstruction to lifting that was articulated in the 1970s, and it answers a question about étale homotopy types negatively.

What carries the argument

Key machinery: the quaternion division algebra D = End(E)⊗Q of a supersingular elliptic curve E. The construction encodes D into the cohomology ring of a blow-up: choose two generating endomorphisms α and β, and consider the five curves given by the two coordinate copies of E, the diagonal, and the graphs of α and β inside (E×E)×P^1. Blowing up each curve at several P^1-points produces exceptional divisor classes; in the quotient of H^2 by products of H^1, these classes are precisely the rank-drop locus of the multiplication map s ↦ s·x. The rank-drop locus is intrinsic to the algebra, and the pairwise distinct cardinalities of the point sets label the five components. Each component's kerne

What would settle it

Search for a graded Q-algebra R_Q and an isomorphism R_Q ⊗_Q Q_ℓ ≅ H^*(X, Q_ℓ) for the X constructed in the paper; finding such an R_Q would directly refute the theorem.

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Extended reading notes

Core claim

The paper proves Theorem 0.1: there is a simply-connected smooth projective variety X over F_p whose étale cohomology ring H^*(X,Q_ℓ) admits no Q-form. A threefold Z is built by blowing up (E×E)×P^1 along graphs of endomorphisms of a supersingular elliptic curve E, using P^1-point sets with distinct cardinalities as labels. The exceptional divisors form exactly the rank-drop locus of a multiplication map, so the ring H^*(Z) intrinsically recovers five subspaces of H^1(Z) that encode the action of the quaternion division algebra D = End(E)⊗Q. A Q-form of H^*(Z) would give a 2-dimensional Q-representation of D, impossible because D is 4-dimensional. Embedding Z in P^9 and blowing up yields a s

Load-bearing premise

The argument that the five subspaces are recoverable requires the blow-up point sets T_γ to be pairwise disjoint with pairwise distinct cardinalities, which needs at least 15 F_p-points on P^1, so the proof as written does not cover primes p < 17.

Editorial extensions

If this is right

  • It supplies a new type of non-liftable variety: the obstruction is in the ℓ-adic cohomology ring itself, not in geometric pathologies like failure of Kodaira vanishing or non-liftable group actions.
  • It realizes a topological obstruction to characteristic-zero lifting that was proposed in the 1970s, and does so for a simply-connected variety.
  • It gives a negative answer to a question about whether étale homotopy types of smooth proper varieties in characteristic p are ℓ-complete equivalent to finite CW complexes.
  • Any smooth proper variety whose cohomology ring has no Q-form is automatically non-liftable, so the construction provides a template for finding more such examples.
  • The method adapts to higher dimensions: embedding Z in any projective space of dimension at least 9 and blowing up gives higher-dimensional examples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The rank-drop technique might be generalizable to encode other finite-dimensional division algebras or even more general algebras into cohomology rings, yielding Q-form obstructions for a broader class of varieties.
  • The distinct-cardinality labeling trick suggests a general recipe: blow up several disjoint loci and use the dimensions of the exceptional blocks as intrinsic labels, which could work in other cohomology theories (e.g., motives) as well.
  • If the finite-field point-count condition is relaxed (e.g., by using a different separation mechanism), the proof might extend to all primes; as written, the argument only covers p ≥ 17.
  • The intrinsic recovery of H^*(Z) from H^*(X) via the hyperplane and exceptional classes may be formalizable as an algorithm that reconstructs a blow-up center from a cohomology ring, with potential applications to other reconstruction problems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper constructs a smooth projective threefold Z over a finite field by blowing up (E×E)×P^1 along graphs of endomorphisms of a supersingular elliptic curve E, with the graphs separated by finite subsets T_γ of P^1, and then forms a simply connected variety X = Bl_Z(P^9). The main theorem asserts that the Q_ℓ-étale cohomology ring of X admits no Q-form, giving a new obstruction to characteristic-zero lifting and a negative answer to Question 0.2. The proof recovers the action of the quaternion algebra D = End(E)⊗Q on H^1(E) from the rank-drop locus in H^2(Z), then argues that a Q-form of H^*(Z) would produce a 2-dimensional Q-vector space representation of a 4-dimensional division algebra, a contradiction. The final blow-up is shown to retain enough of H^*(Z) in its cohomology ring to transfer the obstruction.

Significance. If the construction works, this would be a striking new non-liftability mechanism: a simply connected smooth projective variety whose ℓ-adic cohomology ring is not defined over Q, thereby realizing Sullivan's topological obstruction and giving a negative answer to Grothendieck's Question 0.2. The paper has real strengths: the rank-drop recovery in §2.3 is elegant and the reconstruction of H^*(Z) from H^*(X) in §3.2 is carefully argued and largely self-contained. The argument is not circular and invokes standard theorems (blow-up formula, specialization, quaternion algebra facts). These strengths make the paper potentially publishable, provided the base-field and small-prime gaps identified below are resolved.

major comments (2)
  1. [§2.1 (Construction of Z)] The assertion that End(E)⊗Q is a four-dimensional quaternion division algebra conflates geometric and rational endomorphisms. For a supersingular E/F_p, Silverman's theorem gives End_{\bar F_p}(E)⊗Q as a quaternion algebra, but End_{F_p}(E)⊗Q is the quadratic field Q(π), where π is the p-power Frobenius (π² = -p). Indeed End_{F_p}(E) is contained in the centralizer of π, so any α,β generating the quaternion algebra are not defined over F_p. Hence the graphs Γ_α, Γ_β, the center C, and the blow-up Z are not defined over F_p. Consequently Theorem 0.1's 'over F_p' is not established. Moreover, any F_p-descent would force the Galois action to permute labels, but the pairwise distinct m_γ in Lemma 2.4 rule out nontrivial conjugation of labels; the construction as written appears to yield a variety over F_p² at best. This is load-bearing for the theorem as stated.
  2. [§2.1 (Construction of Z)] The choice of five pairwise disjoint nonempty subsets T_γ⊂P^1(F_p) with pairwise distinct cardinalities m_γ requires at least 1+2+3+4+5=15 points in P^1(F_p). Since #P^1(F_p)=p+1, this forces p≥17. Theorem 0.1 is stated for an arbitrary prime p with no such hypothesis, and the proof gives no alternative construction for small primes. This is an unstated assumption that must be made explicit, or the theorem must be restricted to p≥17.
minor comments (5)
  1. [Lemma 1.1] The same letter K is used for the fraction field of R and for its algebraic closure. Use \bar K for the closure.
  2. [Title/Abstract] The rendering 'NON-LIFT ABLE V ARIETIES' contains spurious spaces; this should be corrected in the final version.
  3. [§2.1] The notation End(E) should be explicitly defined to mean endomorphisms over the algebraic closure if that is intended; but then the paper must address the k-rationality of α and β and of the graphs Γ_γ.
  4. [Proposition 2.2] The sentence 'The same argument as in Lemma 2.4 shows...' is terse. Since the rank-drop recovery over an arbitrary subfield F is used for the Q-form argument, the compatibility with scalar extension deserves a few more words.
  5. [§3.1] The citation [Sha13, Chapter II, §5.4, Theorem 2.25] for embedding a smooth threefold in P^7 should be checked; over finite fields, the standard generic projection argument may require a Bertini-type theorem, so a precise reference or a few words of justification would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained and the no-Q-form obstruction is derived by a genuine contradiction from external theorems.

full rationale

The paper's central claim is not derived from its own conclusion. The construction first chooses a supersingular elliptic curve E and endomorphisms α, β generating the quaternion algebra D, then forms the blow-up Z whose centers are graphs of these endomorphisms. The proof that H*(Z) has no Q-form is a contradiction argument: assuming a Q-form exists, Proposition 2.2 shows the intrinsically recovered subspaces descend to Q, producing a Q-algebra homomorphism ρ: D^op → End_Q(V_Q) with dim_Q V_Q = 2, impossible because D^op is a 4-dimensional division algebra. This is a standard descent-and-ramification contradiction, not a restatement of an input. Proposition 3.3 recovers H*(Z) from H*(X) via the blow-up formula and intrinsically detected classes h and e; the reconstruction is proved by explicit isomorphisms, not by definition of the target. The cited [SZ26] appears only as contextual inspiration and is not load-bearing. There are no fitted parameters, no subsets of data being 'predicted,' and no uniqueness theorem from the authors is invoked to force the construction. The skeptic's concerns—the field of definition of End(E) and the p ≥ 17 cardinality condition for five disjoint T_γ—are mathematical correctness/coverage issues, not circularity, and do not affect the circularity assessment of the derivation chain as written.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The construction has no fitted constants and introduces no new geometric entities. It depends on standard, well-established theorems in étale cohomology, elliptic curves, and blow-up theory. The only hand-chosen combinatorial parameters are the m_γ, which are needed for the label-recovery step and are responsible for the p≥17 restriction.

free parameters (1)
  • m_γ = #T_γ = pairwise distinct positive integers, e.g. 1,2,3,4,5
    Pairwise distinct block dimensions are used in Lemma 2.4 and Proposition 2.2 to identify the five exceptional blocks B_γ and hence the labeled configuration; this requires p+1≥15.
assumptions (8)
  • standard math A supersingular elliptic curve over F_p exists for every p and has rational endomorphism algebra a four-dimensional quaternion division algebra.
    Invoked in §2.1 via [Sil09] to obtain the quaternion algebra D.
  • standard math The étale cohomology blow-up formula and its compatibility with cup products (SGA7/DK73).
    Used throughout §1.2 and §3 to compute H^*(Z) and H^*(X).
  • standard math The specialization isomorphism for smooth proper morphisms respects ring structure (Milne).
    Used in Lemma 1.1 to turn a characteristic-zero lift into a Q-form of the Q_ℓ-cohomology ring.
  • standard math Comparison theorem between singular and étale cohomology after embedding into C.
    Used in Lemma 1.1 to produce a Q-form from a complex model.
  • standard math Étale fundamental group is invariant under blow-up of a smooth center (SGA1).
    Used in §3.1 to conclude that X=Bl_Z(P^9) is simply connected.
  • standard math Extension of scalars from Q to Q_ℓ is faithful.
    Used in Proposition 2.5 to descend relations satisfied after scalar extension back to Q.
  • standard math A smooth projective threefold over F_p embeds in P^7.
    Used in §3.1 via [Sha13] to embed Z before the final blow-up.
  • standard math A four-dimensional division algebra over Q has no two-dimensional representation over Q.
    The final contradiction in Proposition 2.5: a unital hom from D^op to End_Q(V_Q) with dim V_Q=2 is impossible.

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Pith. "Pith review of Non-liftable varieties via etale cohomology rings." pith.science (2026). https://pith.science/paper/B4NTABTI

@misc{pith2026260718588,
  author       = {Pith},
  title        = {Pith review of: Non-liftable varieties via etale cohomology rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B4NTABTI}},
  note         = {Machine review of arXiv:2607.18588}
}
abstract

We construct a smooth projective variety in positive characteristic whose $\mathbb{Q}_{\ell}$-coefficient etale cohomology ring is not the scalar extension of any graded $\mathbb{Q}$-algebra, providing an example of a new type of obstruction to characteristic zero liftability.

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